1-minimal models for $C_{\mathrm{\infty }}$-algebras and flat connections

Given a smooth manifold $M$ equipped with a properly and discontinuous smooth action of a discrete group $G$, the nerve $M_{\bullet }G$ is a simplicial manifold and its vector space of differential forms ${\mathrm{Tot}}_N(A_{DR}(M_{\bullet }G))$ carry a $C_{\mathrm{\infty }}$-algebra structure $m_{\bullet }$. We show that each $C_{\mathrm{\infty }}$-algebra $1$-minimal model $g_{\bullet}: : : \left(W, {m’}{\bullet} \right) \to \left( \operatorname{Tot}{N}\left(A_{DR}(M_{\bullet}G)\right),m_{\bullet}\right) $givesaflatconnection$\nabla$onasmoothtrivialbundle$E$on$M$ where the fiber is the Malcev Lie algebra of ${\pi }_1(M\mathrm{/}G)$ and its monodromy representation is the Malcev completion of ${\pi }_1(M\mathrm{/}G)$. This connection is unique in the sense that different $1$-models give isomorphic connections. In particular, the resulting connections are isomorphic to Chen’s flat connection on $M\mathrm{/}G$. If the action is holomorphic and $g_{\bullet }$ has holomorphic image (with logarithmic singularities), $(\mathrm{\nabla },E)$ is holomorphic and different $1$-models give (holomorphically) isomorphic connections (with logarithmic singularities). These results are the equivariant and holomorphic version of Chen’s theory of flat connections.